Rational and Irrational Number Set proof.

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SUMMARY

The discussion focuses on proving that if \( a \in \mathbf{Q} \) (rational numbers) and \( t \in \mathbf{I} \) (irrational numbers), then both \( a+t \in \mathbf{I} \) and \( at \in \mathbf{I} \). The participants clarify that using proof by contradiction is effective, demonstrating that if \( at \) or \( a+t \) were rational, it would lead to a contradiction with the irrationality of \( t \). The conversation also touches on the properties of sums and products of irrational numbers, concluding that no definitive statement can be made about \( s+t \) or \( st \) for arbitrary irrational numbers \( s \) and \( t \).

PREREQUISITES
  • Understanding of rational (\( \mathbf{Q} \)) and irrational (\( \mathbf{I} \)) numbers
  • Familiarity with proof by contradiction techniques
  • Basic knowledge of algebraic operations involving real numbers
  • Awareness of number sets such as natural (\( \mathbf{N} \)), integers (\( \mathbf{Z} \)), and rational numbers (\( \mathbf{Q} \))
NEXT STEPS
  • Study proof by contradiction in mathematical logic
  • Explore properties of irrational numbers and their operations
  • Learn about the field properties of rational numbers
  • Investigate examples of sums and products of irrational numbers
USEFUL FOR

Mathematicians, students studying number theory, educators teaching properties of rational and irrational numbers, and anyone interested in mathematical proofs and logic.

linuxux
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Hello, here is my problem:

how can i prove that if a\in\mathbf{Q} and t\in\mathbf{I}, then a+t\in\mathbf{I} and at\in\mathbf{I}?

My original thought was to show that neither a+t or at can be belong to N, Z, or Q, thus they must belong to I. However I'm not certain if that train of thought is correct.

Also, i have a question that says given two irrational numbers s and t, what can be said about s+t and st.

My original thought he was that nothing can be shown, since it is possible to create numbers that belong to N, Z, Q, or I.

thanks for clarification.
 
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The rational numbers are a field. Oh, and I is not standard notation, by the way.

As for the second one, then you can't say anythingabout s or t's rationality. Just construct some examples.
 
whoa, thanks, i would have never gotten that.
 
For a moment I thought you were trying to prove that the sum of a rational number and an integer was an integer!
 
The first set of problems are standard proofs by contradiction.

Suppose a is rational and t is irrational and at is rational and a+t is rational.

Since at is rational, at=m/n for appropriate integral m & n.

Then, t=m/na, which is rational. But t is irrational by our hypothesis. Therefore, at cannot be rational, hence it is irrational.

The proof for a+t is similar.
 
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